Hairy-Ball Theorem

I've been getting a lot of questions from readers as to whether the Hairy-Ball Theorem is actually a real thing - or, more to the point, whether its a real thing that is actually useful or if its just an obscure theorem with a funny name. Well, not only is it useful, but its actually quite cool.

Theorem:Hairy-Ball Theorem

Let $n$ be an odd integer and let $S$ be the $n-1$ dimensional unit sphere in $\mathbb{R}^n$. If $V:S \to \mathbb{R}^n$ is a continuous tangent vector field then $V(p) = \mathbf{0}$ for some $p \in S$.

So, that's confusing, right? The theorem talks about even dimensional unit spheres, right? Well, lets just think about the regular two-dimensional sphere. I don't mean two-dimensional as in flat and fits on a piece of paper, I mean the surface of a soccer-ball or something. We can also define it as the set of all points in $\mathbb{R}^3$ such that they are a distance of $1$ away from the origin, or $S = \left \{ \left ( a,b,c \right ) \in \mathbb{R}^3 : \sqrt{a^2 + b^2 + c^2} = 1 \right \}$. Now we let $V:S \to \mathbb{R}^n$ be some continuous tangent vector field. By tangent, I mean that if $p \in S$ is some point on the sphere and $V(p) = \mathbf{v}$ then $\mathbf{v}$ lies tangent to $S$ at the point $p$. So then $V$ places a vector on every point of the sphere such that the vectors are all tangent to the sphere and $V$ must be continuous - the continuity of $V$ is actually very important.

Now, the Hairy-Ball Theorem tells us that no matter how we pick the vectors on the sphere (that is for any continuous tangent vector field, $V$) there has to be a zero vector somewhere on the sphere. That's very cool and not very obvious. As it turns out, if your surface is a doughnut instead of a sphere, its very easy to form a continuous tangent vector field that doesn't have a zero. Then, once you have the Hairy-Ball Theorem, some topology gets you that every two-dimensional differentiable manifold that's homeomorphic to a sphere also has the property that continuous tangent vector fields must vanish somewhere, and there are a lot of two-dimensional differentiable manifolds that are homeomorphic to a sphere.

I kind of glossed over everything, so if you want a little more info on the Hairy-Ball Theorem, the wikipedia page that I put in the references is pretty good. Also, I had every intention of proving it, but in the course of researching it I found someone that had written up basically the exact proof that I would have given, so instead of retyping it I put it in the references. Don't worry if you don't understand it - it takes some real analysis (surprisingly it only takes real analysis) that we haven't touched.

References

Wikipedia: Hairy-Ball Theorem

Topological Musings: Proof of the Hairy-Ball Theorem